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Fredrik

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## Main Question or Discussion Point

What exactly is the Hilbert space of a massive spin 0 particle in non-relativistic QM? The following construction defines a Hilbert space H, but is it the right one? We could e.g use some subspace of H instead. And what if we use Riemann integrals instead of Lebesgue integrals?

Let G be the set of measurable functions [itex]f:\mathbb R^3\rightarrow\mathbb C[/itex] such that [itex]|f|^2[/itex] is Lebesgue integrable. Define an equivalence relation on G by saying that f and g are equivalent if the set on which f-g is non-zero has Lebesgue measure 0, and define H to be the set of such equivalence classes. Define the inner product of two arbitrary equivalence classes [itex]\bar f[/itex] and [itex]\bar g[/itex] by

[tex]\langle \bar f|\bar g\rangle=\int\limits_{-\infty}^\infty d^3x f^*(x) g(x) [/tex]

where f and g are arbitrary functions in the equivalence classes [itex]\bar f[/itex] and [itex]\bar g[/itex].

Let G be the set of measurable functions [itex]f:\mathbb R^3\rightarrow\mathbb C[/itex] such that [itex]|f|^2[/itex] is Lebesgue integrable. Define an equivalence relation on G by saying that f and g are equivalent if the set on which f-g is non-zero has Lebesgue measure 0, and define H to be the set of such equivalence classes. Define the inner product of two arbitrary equivalence classes [itex]\bar f[/itex] and [itex]\bar g[/itex] by

[tex]\langle \bar f|\bar g\rangle=\int\limits_{-\infty}^\infty d^3x f^*(x) g(x) [/tex]

where f and g are arbitrary functions in the equivalence classes [itex]\bar f[/itex] and [itex]\bar g[/itex].